Which of the following inequalities is equivalent to −4 < x < 8?

We know that |x| < a means -a < x < a, where Sum of lower limit of x (i.e -a) and the upper limit of x (i.e a), is 0

Given is, -4 < x < 8, let's say by adding y to this inequality we will get into the above format

-4+y < x+y < 8+y

Now, to move this into the mod format, we need to have (-4+y) + (8+y) = 0 => y = -2

Thus, -6< x-2 < 6 => |x-2| < 6.

Hence, answer is D

The inequality \(|x-a|<b\) is equivalent to inequality \(a- b < x< a+b\) in which a is average of a+b & a-b, and b is mid distance from \(a-b\)and \(a+b.\)

So \(a = {(a+b) + (a-b)}/2\)
\(b= {(a+b) - (a-b)}/2\)

Let's put these values in original equation.

\(a= (-4+8)/2 =2\)

\(b={8-(-4)}/2 = 6\)

\(|x-2|<6\)

Similar question to practice: https://gmatclub.com/forum/which-of-the ... 59334.html
Thanks to MathRevolution for detail explanation.

We see that the first 4 given choices are all in the form of |x - a| < b (for example, |x + 2| < 6 can be considered as |x - (-2)| < 6).

Recall that |x - a| < b (where b is positive) means -b < x - a < b, which becomes a - b < x < a + b. So we want to find a and b such that a - b = -4 and a + b = 8. Adding these two equations, we have: 2a = 4 → a = 2. Substituting 2 for a in a + b = 8, we have b = 6. Therefore, |x - 2| < 6 is the correct inequality.

Answer: D

Question)
Which of the following inequalities is equivalent to −4 < x < 8?
A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above


Solution:

We need to form an inequality with absolute value for the inequality −4 < x < 8
This implies that x will take all values between −4 and 8

The mean of −4 and 8 is 2
Also, the distance of the mean (i.e. 2) from −4 or from 8 is 6
Thus: −4 = (2 − 6) and 8 = (2 + 6)

We know that |x − p | = q implies that the distance of x from p is q units
This results in the values (p − q) and (p + q) for x

Thus, comparing, if we have: |x − 2| = 6, we would get x = 2 − 6 = −4 or x = 2 + 6 = 8
Since we need all the values between −4 and 8 (i.e. we should NOT exceed −4 or 8), we should therefore have:
|x − 2| < 6

Answer D

In this question on inequalities and absolute values, the concept that is being tested is as follows:

If |x-a| < k, then the range of x that satisfies the inequality is given by a-k < x < a+k.

Comparing this with the inequality given in the question, we can say a-k = -4 and a+k = 8. This means a = 2 and k = 6.

This means that -4<x<8 is the range that satisfies the inequality |x-2| < 6. The correct answer option is D.

Hope that helps!

This case is simple use the pattern of absolute values |x − a| = b and then x = a+b and aslo x = a-b

In our case we have a-b = -4 and a+b = 8 so we can try each example to find the correct answer. which is |x − 2| = 6
2 + 6 = 8 and 2-6 = -4